PROJECTIVE CONSTRAINT VIOLATION STABILIZATION METHOD
FOR MULTIBODY SYSTEMS ON MANIFOLDS
Prof. dr. Zdravko Terze
Dept. of Aeronautical Engineering,
Faculty of Mechanical Eng. & Naval Arch.
University of Zagreb
Dr. Joris Naudet
Multibody Mechanics Group
Dept. of Mechanical Engineering
Vrije Universiteit Brussel
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD

Introduction

Focus: constraint gradient projective method for
numerical stabilization of mechanical systems
 holonomic and non-holonomic constraints

Numerical errors along constraint manifold
 optimal partitioning of the generalized
coordinates
 to provide full constraint satisfaction
 while minimizing numerical errors
along manifold
 optimal constraint stabilization effect

Numerical example
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD

Unconstrained MBS on manifolds
- autonomous Lagrangian system, n DOF , n ODE
d  L  L
*
*
,


x, x 
M
x
x

Q





 
dt  x  x

Differentiable-manifold approach:
- configuration space
Rn
 differentiable manifold
M n covered (locally) by coordinate system x (chart)
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD
Mn
Mn
x M
  TxM
 n-dimensional tangent space TxM , x
is not a vector space, at every point :
+ union of all tangent spaces : TM :
 TxM
xM n
 tangent bundle TM n (‘velocity phase space’)
TM   x , x  : x M , x TxM  , dim = 2n
M x   Riemannian metric (positive definite)
TxM
 locally Euclidean vector space
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD


MBS with holonomic constraints
unconstrained system: Mx x  Q* x, x
 , t  , x M
1 2
1
Ek  x Mx   x T Mx x , x  TxM
2
2
- trajectory in the manifold of configuration

holonomic constraints:

T : x i  x i t 
 x, t   0 , Φx, t  : R n R  R
 restrict system configuration space (‘positions’):
n-r dim constraint manifold:
 at the velocity level:
r
T : x i  x i t 
S nr (t )   x  M , x, t   0 
*x x, t x   t  τ
 linear in x
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD

Geometric properties of constraints
*T
- constraint matrix:
 x x, t   1* ,....,*r 
 constraint subspace
r̂1
̂1
M
2

grad Φ1  ˆ 1 .... grad Φ r  ̂ r
Tx M 2
S1
C xr :
 tangent subspace
Tx S n-r :
Tx S n-r  C xr  0 , Tx S n-r  C xr  TxM n
Tx S n-r basis vectors: rˆ1 ,.....,rˆn r   *x (x, t )R(x, t )  0
- constraint submanifold
S nr:
described by y  R n-r
 minimal form formulation
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD

Mathematical model of CMS dynamics
T
 DAE of index 3: Mx x  Q* x , x, t    *x x, t λ
 DAE of index 1:
M
 *
 x
T
*
 *x   x  Q 
    
0  λ   0 
 R TM 
R T Q* 
 *  x  


0
x




 ‘projected ODE’ :
 z
 z , x  R z , x  R z  R
R TM R z  R T Q  R TM R
 integral curve drifts away from submanifold S nr
 only if y  R n-r can be determined that describe S nr
 constraint stabilization procedure is not needed
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD

MBS with non-holonomic constraints

‘r’ holonomic constraints:  x, t   0 

S nr
additional ‘nh’ non-holonomic constraints  x, x , t   0
:
 do not restrict configuration space
/‘positions’
 impose additional constraints on
/‘ velocities’
TS
 x
  Txnr nhS n-r  Txnr S n-r

if linear in velocities (Pfaffian form) ,   B* x, t x  x, t   0
*
 x x, t 

*
x, t R x, t   0

- system constraints  *
,
x


nh



 
 B x, t  
DAE  constraint stabilization procedure
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD

Stabilized CMS time integration

Integration step (DAE or ‘projected’ ODE)
M
 *
 x

T
*
 * x   x  Q 
 λ    
0    ξ 
x ,  ODE

 x x
Stabilization step
 generalized coordinates partitioning:
xd  R r
xi  R n - r
x d  R r
x i  R n - r
 correction of constraint violation  x, t   0 , *x x, t x   t  τ

Problem: inadequate coordinate partitioning
 negative effect on integration accuracy along manifold
 constraints will be satisfied anyway !!
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD

Constraint gradient projective method
 projective criterion to the coordinate partitioning method
(Blajer, Schiehlen 1994, 2003), (Terze et al 2000), (Terze, Naudet 2003)
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD

Questions ?!

If optimal subvector for ‘positions’ is selected:
 is the same subvector optimal choice for
velocity stabilization level as well ?
 is it valid in any case ?

Is the proposed algorithm applicable for
stabilization of non-holonomic systems ?
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD

Structure of partitioned subvectors

System tangent bundle:
TM   x , x  : x M , x TxM  , dim = 2n
TM n  Riemannian manifold , M M  diag Mx , Mx 
T

Holonomic constraints
- ‘position’ constraint manifold
S nr   x  M , x, t   0 
 x correction gradient:
x̂
2
Sn - r
x
1
grad  ( x, t )  0   *x ( x, t )
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD
- velocity constraint manifold
V
nr

 x  TxM , 
*
x
x, t  x  τ 
x̂
V
2
n-r
 x correction gradient :
x
1
grad  *x x, t x     *x ( x, t )
 Holonimic systems: optimal partitioning returns ‘the same
dependent coordinates’ at the position and velocity level
grad  x, t   0   *x ( x, t )  grad  *x x, t  x  
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD

Non-holonomic constraints

linear (Pfaffian form):
  B* x, t x  x, t   0
H + NH constraints:
 *x x, t 

*
 *
 x   nh x  
 
 B x, t  

 x correction gradient:
 *
  
grad  nh x       *nh ( x, t )
 

 x correction gradient:
grad  ( x, t )  0   *x ( x, t )
 Non-holonomic systems: correction gradients do not
match any more. A separate partitioning procedure
for stabilization at configuration and velocity level !!
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD

Coordinates relative projections vs time
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD

Non-holonomic mechanical system
- dynamic simulation of the satelite motion (INTELSAT V)
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD

Reference trajectories
DEPARTMENT OF AERONAUTICAL ENGINEERING
CHAIR OF FLIGHT VEHICLE DYNAMICS
CONSTRAINT GRADIENT PROJECTIVE METHOD

Relative length of projections on constraint subspace